PARTHENON MATH AND GREAT PYRAMID
L. Kaliambos (Natural Philosopher) February 14 , 2016 A detailed comparison of the dimensions of the Great Pyramid at Giza with the dimensions of the Parthenon led to my discovery of math about the plan of the construction of columns of Parthenon which form a theoretical pyramid of very large height H = 1811.3 m. It is well known that from the time of Thales of Miletus who calculated the initial height h = 146.5 m of the Cheops pyramid Greeks were aware about the math and the dimensions of pyramid.(Great Pyramid of Giza-Wikipedia) This photo is from the intrview I gave with the title "Parthenon math"(Μαθηματικά του Παρθενώνα) to the author of Spiritual Thessaly, Mrs. Dimitra Bardani through the TV Thessalia (Greece). '' According to “Thales-WIKIPEDIA” Thales (624- 546 BC) was known for his innovative use of geometry. He understood similar triangles and right triangles, and measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid’s shadow measured from the center of the pyramid at that moment must have been equal to its height. So it means that Greek mathematicians from the time of Thales were aware about the dimensions and the mathematics of pyramids. In my paper “Discovery of Parthenon math” I showed that the mathematics of the golden section was used not only for the construction of the Cheops pyramid (2560 BC) but also in Parthenon and in Hephaestion tomb. Therefore the math of golden section was well known to Phidias, and Dinocrates. Nevertheless, for understanding better the use of golden section based on golden rectangles in Parthenon built by the ancient Greeks from 447 to 438 BC you can see in Google “The Parthenon and Phi, the Golden Ratio”. In fact, in my paper “Discovery of Φ and π in Giza great pyramid ” I showed that both the Pi = π = C/d =3.14... and the Phi = Φ = (1+50.5)/2 = 1.618... were used in the construction of Cheops pyramid . Golden Section Phi = Φ = (1 + 50.5)/2 is obtained by dividing a line into two parts (α and β) such that the square of the first part is equal to the product of the whole segment (α+β) and the second part. That is α2 = (α +β )β or (α +β )/α = α/β = Φ One method for finding the value of Φ is to write α = Φ and β = 1 (unit length). Therefore (Φ+1)/Φ = Φ/1 or Φ + 1 = Φ2 which can be rearranged to Φ2 - Φ -1 = 0. Today it is well known that in a quadratic formula of the following general form aΦ2 + bΦ + c = 0 the value (positive ) of Φ is given by Φ = [ -b + (b2 -4ac)0.5] / 2a Since a = 1, b = -1, and c = -1 one gets Φ = (1+ (1+4)0.5] / 2 = (1+50.5) / 2 = 1.6180339887…. Such a golden number is obvious in my discovery of the golden section in Caryatids of Erechtheion. The Egyptians used the Phi = Φ = 1.618…) in the design of the Great Pyramids and they thought that the golden ratio was sacred. Therefore, they used the golden ratio when building temples and places for the dead. The Egyptians were aware that they were using the golden ratio Φ. However, one should ask how the ancient Egyptians were able to find the solution of the quadratic formula Φ2 - Φ -1 = 0 . The study of this called algebra goes to the antiquity. Recent discoveries have shown that Babylonians and Egyptians solved problems in algebra , although they had no symbols for variables. They used only words to indicate such numbers, and for that reason their algebra has been referred to as theoretical algebra. The Ahmes Papyrus, an Egyptian scroll going back to 1600 BC has a number of problems in algebra, in which the unknown is referred to as a hau, meaning “a heap”. Also practically the so -called Pythagorean theorem (6thcentury BC) was well known to Babylonians and Egyptians. Thus writing 1 + Φ = Φ2 as (1)2 + (Φ0.5)2 = Φ2 one sees that the 1 (unit length) should be the radius r of a cone pyramid, while Φ0.5 = h (height) and Φ = L (slant height). In this case the circumference C =2π because r = 1. In other words such a cone pyramid was believed to be a sacred pyramid because it includes the mystic numbers Phi = Φ and Pi = π . Also the great pyramid of Cheops was believed to be a sacred square pyramid, because a theoretical cone pyramid was inscribed in the square pyramid. Another ratio does appear throughout most of the Parthenon. This equals a 9 : 4 ratio which can be related to the sacred Pythagorean rectangles of the numbers 3, 4, and 5. According to the history of Greek People (Volume Γ2 page 282) the dimensions of the Parthenon are Height z = 13.724 m . Width y = 30.88 m. Length x = 69.48 m In general for the construction of the Temple of Athena Parthenos the mathematical principles of proportion of the Pythagoreans were used like the basic rectangle of sides 3 and 4 giving the diagonal of 5 as 32 + 42 = 52 Also a rectangle of sides 9 : 4 was constructed from three rectangles of sides 3 and 4 with diagonal 5. In this case the total ratio is given by 9 : 4 = (3/4 + 3/4 + 3/4) Here the ratios 9 : 4 = 2.25 and 92 : 42 = 5.0625 of three Pythagorean rectangles were fundamental to the construction. In other words a basic rectangle of sides 9 : 4 was constructed from three rectangles of sides 3 and 4 with diagonal 5. Under this condition we write y/z = 30.88/13.724 = 2.25 = 9/4 or y = (9/4)z x/y = 69.48/30.88 = 2.25 = 9/4 or x =(9/4)y x/z = 69.48/13.724 = 5.0625 = 92/42 or x = (92/42)z = (5.0625)z Therefore a construction based on the golden numbers 1.618 and 2.618 cannot give impressive dimensions. Whereas the use of the Pythagorian rectangles give the Parthenon structure under a perfect harmony. Now let us calculate the height H of the theoretical pyramid existing over the temple along a vertical direction toward to the sky. According to the "GREEK SURNAMES: Τα μαθηματικά του Παρθενώνα” The columns in Parthenon actually lean slightly inwards so that if they carried on they would make a very tall theoretical pyramid with a volume Vt = Vc /2 where Vt is the volume of the theoretical pyramid of Parthenon and Vc is the volume of the Cheops pyramid. Under this condition we write Vt = xyH/3 = Vc / 2 = α2h/6 or xyH = α2h/2 Here y = 30.88 m is the width of the Parthenon and x = (9/4)y is the length . On the other hand α is the side of the square of the Great pyramid and h is the height h = αΦ0.5/2. Under this condition we may write y2H = (4/9)(α3Φ0.5/4) or H = (4/9)(Φ0.5/4)( α3/y2) or H = (Φ0.5 α3) / (3y)2 Here we see that the height H of the theoretical pyramid of Parthenon (438 BC) depends on the dimension y of Parthenon and the side α of the Cheops square pyramid (2560 BC) including the Pythagorean number 3 , and the golden number Φ =(1 + 50.5)/2 = 1.618.. That is, the vertical line H existing over the head of the goddess of the wisdom should contain all knowledge of mathematics from Cheops to Thales and Pythagoras. Now let us find the height H of the theoretical pyramid of Parthonon by equating its volume with the half volume of Cheops pyramid. According to Wikipedia ” today each side of the pyramid has a length α = 230.34 m. So under the mathematics of golden section the height h is given by h = αΦ0.5/2 = 146.5 m. Since y = 30.88 m and α = 230.34 m we get H = (Φ0.5 α3) / (3y)2 = 1811.3 m To conclude we see that in Parthenon there are many geometric constructions of the Golden Ratio based on a golden rectangle whose ratio of the longer side to the shorter side is Φ = (1 + 50.5)/2 = 1.618... Also in Caryatids of Acropolis I discovered that the method of designing the golden section is the same as that of the Caryatids in Hephaestion tomb. Nevertheless for the design of the whole temple having impressive dimensions under a harmony Ictinus and Callicrates decided to construct it by using the mathematics of Pythagorean rectangles with the ratio 9 : 4. Moreover in order to show that the height H over the goddess of wisdom symbolizes all knowledge of math from the time of Cheops to the time of Thales and Pythagoras Ictinus and Callicrates used the Pythagorean number 3 along with the golden number Φ = (1 + 50.5) /2 for determining the height H after a careful comparison of the dimensions of the Cheops pyramid with the dimensions of Parthenon. Category:Fundamental physics concepts